Based on semi-direct sums of Lie subalgebra G, a higher-dimensional 6 × 6 matrix Lie algebra sμ（6） is constructed. A hierarchy of integrable coupling KdV equation with three potentials is proposed, which is de...Based on semi-direct sums of Lie subalgebra G, a higher-dimensional 6 × 6 matrix Lie algebra sμ（6） is constructed. A hierarchy of integrable coupling KdV equation with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, the Hamiltonian forms is deduced for lattice equation in the resulting hierarchy by means of the discrete variational identity -- a generalized trace identity. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian systems.展开更多
In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions ar...In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.展开更多
The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With th...The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.展开更多
It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons i...It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons in a semidiserete integrable system （the time variable is continuous and the space one is discrete） that the shorter solitary waves travel faster than the taller ones. Very recently, this kind of soliton was also observed in a full discrete generalized KdV system （the both of time and space variables are discrete） introduced by Kanki et al. In this paper, for the generalized discrete KdV （gdKdV） equation, we describe its richer structures of one-soliton solutions. The interactions of two-soliton waves to the gdKdV equation are studied. Some new features of the soliton interactions are proposed by rigorous theoretical analysis.展开更多
We derive bilinear forms and Casoratian solutions for two semi-discrete potential Korteweg-de Vries equations. Their continuum limits go to the counterparts of the continuous potential Korteweg-de Vries equation.
基金Project Supported by National Nature Science Foundation of China(10461006)the High Education Science ResearchProgramof Inner Mongolia(NJ02035)the Natural Science Foundation of Inner Mongolia(2004080201103)
基金Supported by the Nature Science Foundation of Shandong Province of China under Grant No.ZR.2009GM005the Science and Technology Plan Project of the Educational Department of Shandong Province of China under Grant No.J09LA54the research project of "SUST Spring Bud" of Shandong University of Science and Technology of China under Grant No.2009AZZ071
文摘Based on semi-direct sums of Lie subalgebra G, a higher-dimensional 6 × 6 matrix Lie algebra sμ（6） is constructed. A hierarchy of integrable coupling KdV equation with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, the Hamiltonian forms is deduced for lattice equation in the resulting hierarchy by means of the discrete variational identity -- a generalized trace identity. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian systems.
基金The project supported by the State Key Basic Research Program of China under Grant No 2004CB318000
文摘In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.
基金Supported by the National Natural Science Foundation of China under Grant No. 60772023by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. BUAA-SKLSDE-09KF-04+2 种基金Beijing University of Aeronautics and Astronautics, by the National Basic Research Program of China (973 Program) under Grant No. 2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 200800130006Chinese Ministry of Education, and Scientific Research Common Program of Beijing Municipal Commission of Education under Grant No. KM201010772020
文摘The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11501353,11271254,11428102,and 11671255supported by the Ministry of Economy and Competitiveness of Spain under contracts MTM2012-37070 and MTM2016-80276-P(AEI/FEDER,EU)
文摘It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons in a semidiserete integrable system （the time variable is continuous and the space one is discrete） that the shorter solitary waves travel faster than the taller ones. Very recently, this kind of soliton was also observed in a full discrete generalized KdV system （the both of time and space variables are discrete） introduced by Kanki et al. In this paper, for the generalized discrete KdV （gdKdV） equation, we describe its richer structures of one-soliton solutions. The interactions of two-soliton waves to the gdKdV equation are studied. Some new features of the soliton interactions are proposed by rigorous theoretical analysis.
基金Supported by National Natural Science Foundation of China under Grant No.11371241National Natural Science Foundation of Shanghai under Grant No.13ZR1416700
文摘We derive bilinear forms and Casoratian solutions for two semi-discrete potential Korteweg-de Vries equations. Their continuum limits go to the counterparts of the continuous potential Korteweg-de Vries equation.
